Sr Examen
Integral de (3*x-5)/(x^2+4*x+6) dx
Solución
Ha introducido
[src]
1 / | | 3*x - 5 | ------------ dx | 2 | x + 4*x + 6 | / 0
$$\int\limits_{0}^{1} \frac{3 x - 5}{\left(x^{2} + 4 x\right) + 6}\, dx$$
Integral((3*x - 5)/(x^2 + 4*x + 6), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 3*x - 5 | ------------ dx | 2 | x + 4*x + 6 | /
Reescribimos la función subintegral
2*x + 4
3*------------ /-11 \
2 |----|
3*x - 5 x + 4*x + 6 \ 2 /
------------ = -------------- + ------------------------
2 2 2
x + 4*x + 6 / ___ \
|-\/ 2 ___|
|-------*x - \/ 2 | + 1
\ 2 / o
/ | | 3*x - 5 | ------------ dx | 2 = | x + 4*x + 6 | /
/
|
| 1
11* | ------------------------ dx /
| 2 |
| / ___ \ | 2*x + 4
| |-\/ 2 ___| 3* | ------------ dx
| |-------*x - \/ 2 | + 1 | 2
| \ 2 / | x + 4*x + 6
| |
/ /
- --------------------------------- + --------------------
2 2 En integral
/
|
| 2*x + 4
3* | ------------ dx
| 2
| x + 4*x + 6
|
/
--------------------
2 hacemos el cambio
2 u = x + 4*x
entonces
integral =
/
|
| 1
3* | ----- du
| 6 + u
|
/ 3*log(6 + u)
------------- = ------------
2 2 hacemos cambio inverso
/
|
| 2*x + 4
3* | ------------ dx
| 2
| x + 4*x + 6
| / 2 \
/ 3*log\6 + x + 4*x/
-------------------- = -------------------
2 2 En integral
/
|
| 1
-11* | ------------------------ dx
| 2
| / ___ \
| |-\/ 2 ___|
| |-------*x - \/ 2 | + 1
| \ 2 /
|
/
----------------------------------
2 hacemos el cambio
___
___ x*\/ 2
v = - \/ 2 - -------
2 entonces
integral =
/
|
| 1
-11* | ------ dv
| 2
| 1 + v
|
/ -11*atan(v)
---------------- = -----------
2 2 hacemos cambio inverso
/
|
| 1
-11* | ------------------------ dx
| 2
| / ___ \
| |-\/ 2 ___|
| |-------*x - \/ 2 | + 1 / ___\
| \ 2 / ___ | ___ x*\/ 2 |
| -11*\/ 2 *atan|\/ 2 + -------|
/ \ 2 /
---------------------------------- = -------------------------------
2 2 La solución:
/ ___\
___ | ___ x*\/ 2 |
/ 2 \ 11*\/ 2 *atan|\/ 2 + -------|
3*log\6 + x + 4*x/ \ 2 /
C + ------------------- - ------------------------------
2 2
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |\/ 2 *(2 + x)| | / 2 \ 11*\/ 2 *atan|-------------| | 3*x - 5 3*log\6 + x + 4*x/ \ 2 / | ------------ dx = C + ------------------- - ---------------------------- | 2 2 2 | x + 4*x + 6 | /
$$\int \frac{3 x - 5}{\left(x^{2} + 4 x\right) + 6}\, dx = C + \frac{3 \log{\left(x^{2} + 4 x + 6 \right)}}{2} - \frac{11 \sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} \left(x + 2\right)}{2} \right)}}{2}$$
Gráfica
Respuesta
[src]
/ ___\
___ |3*\/ 2 |
11*\/ 2 *atan|-------| ___ / ___\
3*log(6) 3*log(11) \ 2 / 11*\/ 2 *atan\\/ 2 /
- -------- + --------- - ---------------------- + --------------------
2 2 2 2
$$- \frac{11 \sqrt{2} \operatorname{atan}{\left(\frac{3 \sqrt{2}}{2} \right)}}{2} - \frac{3 \log{\left(6 \right)}}{2} + \frac{3 \log{\left(11 \right)}}{2} + \frac{11 \sqrt{2} \operatorname{atan}{\left(\sqrt{2} \right)}}{2}$$
=
=
/ ___\
___ |3*\/ 2 |
11*\/ 2 *atan|-------| ___ / ___\
3*log(6) 3*log(11) \ 2 / 11*\/ 2 *atan\\/ 2 /
- -------- + --------- - ---------------------- + --------------------
2 2 2 2
$$- \frac{11 \sqrt{2} \operatorname{atan}{\left(\frac{3 \sqrt{2}}{2} \right)}}{2} - \frac{3 \log{\left(6 \right)}}{2} + \frac{3 \log{\left(11 \right)}}{2} + \frac{11 \sqrt{2} \operatorname{atan}{\left(\sqrt{2} \right)}}{2}$$
-3*log(6)/2 + 3*log(11)/2 - 11*sqrt(2)*atan(3*sqrt(2)/2)/2 + 11*sqrt(2)*atan(sqrt(2))/2
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.